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A connective differential graded C∞-ring is a (homologically graded with nonnegative degrees) real commutative differential graded algebra A equipped with a structure of a C^∞-ring on A0.
A coconnective differential graded C∞-ring is a (cohomologically graded with nonnegative degrees) real commutative differential graded algebra A equipped with a structure of a C^∞-ring on A0 such that the differential d:A0→A1 in degree 0 is a C^∞-derivation.
In the unbounded case, Carchedi–Roytenberg proposed the following definition:
An unbounded differential graded C∞-ring is a (homologically graded with arbitrary degrees) real commutative differential graded algebra A equipped with a structure of a C^∞-ring on Ac0=ker(A0→A−1).
\begin{remark} Every coconnective differential graded C^∞-ring in the sense defined above is also an unbounded differential graded C^∞-ring concentrated in nonpositive homological degrees, since the kernel of a C^∞-derivation is a C^∞-ring. The converse is false: an unbounded differential graded C^∞-ring concentrated in nonpositive homological degrees has a C^∞-ring structure on its 0-cocycles only, which is not enough to reconstruct a C^∞-structure on the whole degree 0 part or ensure that the degree 0 differential is a C^∞-derivation. The stronger condition is essential for some theorems about coconnective differential graded C^∞-rings, such as the one that states that smooth differential forms form the free C^∞-DGA on smooth functions. \end{remark}
Differential graded C^∞-rings can be equipped with the model structure transferred from the projective model structure on chain complexes via the forgetful functor.
Restricting to connective differential graded C^∞-rings, the resulting model structure is Quillen equivalent to the model category of simplicial C^∞-rings equipped with the model structure transferred along the forgetful functor to simplicial sets equipped with the Kan–Quillen model structure. The right adjoint functor is the normalized chains functor, which sends a simplicial C^∞-rings to its normalized chains equipped with the induced structure of a differential graded C^∞-ring. This is analogous to the monoidal Dold–Kan correspondence.
In this form, the statement was first proved in Taroyan 2023. Similar, but not entirely equivalent results can be found in Nuiten 2018, Remark 2.2.11, which uses a homotopy coherent variant of C^∞-rings and does not explicitly identify the adjoint functors.
The essential ingredients (C^∞-Kähler differentials and C^∞-derivations) appear in
The earliest known occurrence of differential graded C^∞-rings is in the paper
where at the bottom of page 28 in arXiv version 1 one reads:
The underlying algebra in degree 0 can be generalized to an algebra over some Lawvere theory. In particular in a proper setup of higher differential geometry, we would demand CE(𝔤)0 to be equipped with the structure of a C^∞-ring.
Additional references:
On unbounded differential graded rings for arbitrary Fermat theories (including C^∞-rings):
On the equivalence of connective differential graded C^∞-rings and simplicial C^∞-rings via the normalized chains functor:
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